Answer by Professional Japanese Teacher
Usually in a Japanese conversation, you do not have to spell out the whole sentence if the context is very clear to the speaker and listener. In your example, it is clear that your team would fall behind in the schedule if the other team does not deliver on time.
In fact, Colleague A’s sentence actually means the other team has to deliver their work now “そろそろ送(おく)ってくださいませんか Sorosoro okuttekudasaimasenka (Can you send it soon?)”.
Of course, Colleague A can also complete his sentence, such as “そろそろ送(おく)ってもらわないと、こちらはスケジュールに遅(おく)れますよ Sorosoro okutte morawanaito, kochira wa sukejuuru ni okuremasu yo (if you don’t send it to us soon, we will fall behind the schedule)”.
However, this sentence would sound too reprimanding to Colleague B from the other team. Hence, leaving sentences incomplete or with no conclusion is a way of being considerate or mindful of the others’ situation.